All assemblies which had a partial solution of level-8 or higher were saved for later, more thorough analysis. Three different programs were run on each assembly:
The results of these analyses are summarized in a file referred to as an 'AF Format' file.
The goal of the analysis programs was to find different, high-level methods for disassembling 6-piece burrs. In most assemblies, minor changes can be made to the internal cube arrangements which do not change the way in which the first piece is removed. Usually there are large groups of assemblies which come apart in virtually identical fashions. A shorthand notation for the disassembly method was developed so that comparisons could easily be made between different assemblies to see if they functioned in the same fashion. This shorthand notation is called the "Apart-Code".
For example, the L5 Notchable puzzle has a level-5 solution. The LL format of the assembly found by the program was:
55551133140220226630440060666606 0 7 2 5 0 0 0The Apart-Code for this solution is:
3146 1124 316 11246This code details the first 4 moves of the disassembly. During the 5th move, the pieces come apart, and this move is not included. The interpretation is as follows:
Move1 Move2 Move3 Move4 xuppp xuppp xuppp xuppp 3146 1124 316 11246Each move is represented by from 3 to 5 characters. The first character ('x') denotes the direction of the move: either 1, 2 or 3. The second character ('u') denotes the number of units that the pieces move in this direction: 1,2,3 for movements in the positive direction and F,E,D, respectively, for 1,2,3 units in the negative direction. Finally, the 3rd and subsequent characters ('ppp') denote the piece numbers which are moved. (up to 3 pieces can be moved at once).
Thus, the moves represented by the above apart-code are:
Move1: In direction #3, move pieces 4 & 6 +1 unit Move2: In direction #1, move pieces 2 & 4 +1 unit Move3: In direction #3, move piece 6 +1 unit Move4: In direction #1, move pieces 2, 4 & 6 +1 unit
Many assemblies have more than one way to arrive at the first disassembly in the same number of moves. This is true of the L5 Notchable puzzle as well. It has 4 different ways to make the first 4 moves, after which a disassembly is possible in one move. The Apart-Codes for these movements are:
3146 1124 316 11246 3146 1124 316 1F13 3146 11245 316 11246 3146 11245 316 1F13
It was necessary to choose one of these as the 'standard' solution. Rules were made to decide between any two moves, or a group of moves, as to which was the 'standard' move. Generally speaking, the rules are:
When the rules are applied to choose one solution from a group of 2 or more, the move(s) which differ between the solutions are marked at the end with a '*'. The 'standard' solution from the above four solutions is:
3146 1124 * 316 1F13 *
There remains one other significant problem to resolve. Each assembly has 24 rotations and/or reflections which are usually different, and will produce 24 different apart-codes. Many assemblies which are similar to a given assembly will be analyzed with the same rotation as the given assembly, but others will be analyzed in a different rotation, particularly those which involve cube changes to the first piece. The answer to this problem is to compare the solutions and related Apart-Codes in all of the 24 rotations and reflections, and choose the 'best' or 'standard' one. The rules given above are used extensively for this. The resulting standard solution for the L5 Notchable includes not only an Apart-Code, but the LL format of the rotation chosen from the 24. The LL format of the rotation chosen is:
50501135442002201133442622226060and the standard Apart-Code is:
1126 2146 * 112 2F35 *
The information from the three analyses, including standardization of the Apart-Code, is combined and saved in one line per assembly in files which are called 'AF Format' files. The fields in these files are described below. In order to conserve space, many of the fields which contain a value from 0 to 12 (or possibly higher) are restricted to a one-character field. In these fields, the letters 'A', 'B', 'C', etc. are used to denote values of 10,11,12, respectively.